Linear Scheduling Model with Varying Production Rates...new linear scheduling model, a linear...

Linear Scheduling Model with Varying Production RatesGregory A. Duffy1; Garold D. Oberlender2; and David Hyung Seok Jeong, A.M.ASCE3

Abstract: Activity production rates drive the development and accuracy of linear schedules. The nature of linear projects dictates an assort-ment of variables that affect each activity’s production rate. The purpose of this research was to expand the capabilities of linear scheduling toaccount for variance in production rates when and where the variance occurs and to enhance the visual capabilities of linear scheduling. Thisnew linear scheduling model, a linear scheduling model with varying production rates (LSMVPR), has two objectives. The first is to outline aframework to apply changes in production rates when and where they occur along the horizontal alignment of the project. The secondobjective is to illustrate the difficulty or ease of construction through the time-location chart. This research showed that the changes inproduction rates because of time and location can be modeled for use in predicting future construction projects. Using the concept of workingwindows, LSMVPR allows the scheduler to develop schedule durations on the basis of minimal project information. The model also allows thescheduler to analyze the impact of various routes or start dates for construction and the corresponding impact on the schedule. The graphicalformat allows the construction team to visualize the obstacles in the project when and where they occur by using a new feature called theactivity performance index (API). This index is used to shade the linear scheduling chart by time and location with the variation in colorindicating the variance in predicted production rate from the desired production rate. DOI: 10.1061/(ASCE)CO.1943-7862.0000320.© 2011 American Society of Civil Engineers.

CE Database subject headings: Project management; Scheduling; Productivity.

Author keywords: Project scheduling; Linear projects; Linear scheduling method; Production rate.

Introduction

Linear construction projects, such as oil and gas pipeline construc-tion, involve continuous, linear activities performed along the hori-zontal alignment of the facility. Although bar charts and the criticalpath method (CPM) are currently the most common schedulingmethods of these types of projects, these methods lack detail whenscheduling linear projects. In linear projects, typically the samecrew repeats each of the activities from one end of the projectto the other. Often the only distinguishing feature for these linear-type activities is their rate of progress. For instance, in pipeline con-struction projects, the sequence of activities is usually not the issueof concern; instead, the issue is accurately assessing and achievingthe optimum production rates necessary for timely completion.Thus, to effectively schedule linear construction, it is necessaryto focus on repetitive-work activities and the probable productionrates rather than the interrelationships among activities. Linear(time-location) scheduling is a technique that better depicts linearactivities than bar charts and CPM in this context, and thus has thepotential to enhance the scheduling of linear projects.

Although linear scheduling has been in existence for quite sometime, its use in the United States construction industry has been

very limited, compared to bar charts and CPM. The primary reasonfor the lack of widespread use of linear scheduling is the lack ofcommercially available software in the United States that addressesthe industry’s needs. In addition, aggressive marketing by CPMsoftware developers has also helped CPM dominate the U.S.market and diminished the use of other scheduling techniques.

The power of the linear scheduling method does not lie in itsability to organize a project’s individual activities, but instead itis gained from the multitude of graphical capabilities inherent tothis method. The use of graphics and the visual intuitiveness pro-vided by the separate activity types enables project managers,schedulers, owners, and construction personnel to better visualizethe plan of action and more easily communicate the plan to every-one involved with the project. Although much research has beenperformed to predict the production rate based on simulation, prob-ability, or regression analysis (Chao and Skibniewski 1994; Smith1999; Kuo 2004; O’Connor and Huh 2005; Chong 2005; Jiang andWu 2007), no significant research has been performed to determinewhen and where the production rates change along the project’salignment. A linear schedule consists of a chart with location orstationing along the one axis and time on the other axis. Thistime-location chart provides the perfect canvas for depicting thechange of production rates when and where they occur.

The purpose of this paper is to present a framework for linearscheduling that accounts for variance in production rates when andwhere the variance occurs and to enhance the visual capabilities oflinear scheduling. The framework can be supported by empiricallyderived production equations with the appropriate variables input atthe appropriate time and location in the project. For example, pro-duction rates of ditching across flat prairie will greatly exceed thatof ditching through mountainous terrain. A given project may con-sist of both types of terrain; therefore, using one production rate oran average production rate would lead to erroneous expectations inthe two unique areas. It would therefore be more useful to applyproduction variables at the appropriate changes in conditions.

1Former Doctoral Student; Engineering Supervisor, Chesapeake Mid-stream Partners, Oklahoma City, OK 73118. E-mail: [email protected]

2Professor Emeritus, School of Civil and Environmental Engineering,Oklahoma State Univ., Stillwater, OK 74074. E-mail: [email protected]

3Associate Professor, School of Civil and Environmental Engineering,Oklahoma State Univ., Stillwater, OK 74074 (corresponding author).E-mail: [email protected]

Note. This manuscript was submitted on January 10, 2010; approved onNovember 12, 2010; published online on November 16, 2010. Discussionperiod open until January 1, 2012; separate discussions must be submittedfor individual papers. This paper is part of the Journal of ConstructionEngineering and Management, Vol. 137, No. 8, August 1, 2011. ©ASCE,ISSN 0733-9364/2011/8-574–582/$25.00.

574 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / AUGUST 2011

J. Constr. Eng. Manage. 2011.137:574-582.

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http://dx.doi.org/10.1061/(ASCE)CO.1943-7862.0000320




These different production variables would in turn be applied to theproduction rate of the activity as it moves through the given areaand time window. This allows the project team to better understandhow and when the production variables affect the constructionprogress throughout the length of the project. It may be possibleto bypass certain drops in production performance simply byunderstanding the compound effect of the production variables.

Previous Research

Two distinct forms of linear scheduling have emerged: repetitivescheduling, referred to in this paper as point-based scheduling, andlocation-based (alignment-based) scheduling. Examples of point-based projects include multiunit housing complexes and high-risebuildings, whereas examples of location-based (alignment-based)projects include pipelines, railways, and highway constructionprojects. The focus of this paper is on those projects classified aslocation-based linear projects. Within the framework of location-based linear scheduling, much research has been performed withvarying nomenclature (Johnston 1981; Chrzanowski and Johnston1986; Voster et al. 1992; Harmelink 1995; Mattila 1997; Harmelinkand Rowings 1998; El-Sayegh 1998; Liu 1999; Herbsman 1999;Yamin 2001; Cosma 2003; Yen 2005). The following sectionbriefly summarizes the advancements made in the linear schedulingmethod for location-based construction projects.

Johnston (1981) introduced the term linear scheduling method(LSM) to the highway construction industry. Johnston used produc-tion rates, activity interruptions, buffers, calendar considerations,and project resources to develop linear schedules for highway con-struction projects. Chrzanowski and Johnston (1986) contrastedCPM with LSM by using an as-built highway schedule. The sim-plicity of LSMwas noted as its largest asset. However, there may betimes when it would be advantageous to use LSM in conjunctionwith CPM. The authors noted that the user “receives fairly detailedinformation without being confronted with the numerical data and

degree of abstraction found in network methods.” They also ad-dressed some of the limitations of linear scheduling. For a projectwith discrete activities, a network diagram may be needed to modelthe interrelationships and sequencing of activities. If a project hasmultiple alignments, such as two intersecting roadways, then it maybe necessary to develop a separate schedule for each roadway,which would require multiple schedules for a single project. In con-clusion, the authors noted that LSM was best used as a complementto CPM.

Harmelink (1995) developed a model of linear scheduling inconjunction with an AutoCAD-based program. His work focusedon two important aspects of linear scheduling: (1) proving comput-erization of linear scheduling is possible and (2) illustrating proce-dures to identify the controlling activity path in the schedule. InCPM, the critical path is defined as the longest path, time-wise,through the sequence of activities. In LSM, an analogous path iscalled the controlling activity path.

As shown in Fig. 1, Harmelink used three key features to definethe controlling activity path. These key features are the least timeinterval (LT), coincident duration, and the least distance interval(LD). The least time interval is “the shortest time interval betweenany two consecutive activities.” The coincident duration is “an in-terval in time during which the two activities connected by the leasttime interval are both in progress.” Lastly, the least distance intervalis “the shortest distance between any two activities that lies withinthe coincident duration interval and intersects the least timeinterval.” The LT, coincident duration, and LD for the pavingand striping activities are depicted in Fig. 1. The coincident dura-tion between weeks 7 and 9 (highlighted in yellow in Fig. 1) showsthe LT and LD interrelationship between the activities of pavingand striping and signage. Another coincident duration exists be-tween weeks 4 and 5 because of the LT and LD interrelationshipbetween the activities of grading and paving; however, this coinci-dent duration is not highlighted to prevent excessive detail in Fig. 1.

Fig. 1. Example of a linear schedule with the controlling activity path displayed

JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / AUGUST 2011 / 575


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El-Sayegh (1998) developed deterministic and probabilisticmodels for calculating resource-based linear schedules. The deter-ministic model can be used to produce a linear schedule basedsolely on user input. The probabilistic model may be used to pro-duce a linear schedule based on Monte Carlo simulation, whichaccounts for variability and uncertainty of construction projects.The models were included in a Windows-based software packagenamed Linear Construction Planning Model (LCPM).

Yamin (2001) developed an approach to analyze the cumulativeeffect of productivity rate variability (CEPRV) on linear activities inhighway projects. The focus of the research was to advance the riskanalysis capabilities of linear scheduling to allow mangers to fore-cast the probability of project delay. This and other statistical analy-sis tools are prevalent with CPM, but they are lacking in linearscheduling methods. Yamin also developed methods for determin-ing secondary controlling activity paths (SCAPs). These SCAPsoccur because of activities that are near critical and have high pro-ductivity rate variability (PRV). The probability that such activitiesmay become critical is high. The author suggests further research inevaluating PRV by statistically analyzing construction factors, suchas type of work being done, soil conditions, weather, equipmenttype, experience of labor, and general layout. This would enablemanagers and schedulers to better forecast the impacts of the vari-ability of the different components.

Production Variables

Production variables are variables that can affect the productionrates of the construction activities. Although many variablesmay influence the actual production rates achieved in the field, theycan be separated into four types:1. General variables. Broad constraints that affect the production

but are not related to a specific time or location.2. Time variables. Variables that change with respect to time only.3. Location variables. Variables that change with respect to loca-

tion only.4. Time-location variables. Variables that change with respect to

both time and location.Table 1 depicts the four types of production variables with exam-ples of common variables in each category. The next section elab-orates on specific variables that affect production rates in each ofthe four categories.

General production variables by definition do not change withrespect to time or location. An example of this is the number ofworkers on the project, which is typically a constraint set by theproject team and/or the current market demand and/or the availabil-ity for that type of labor. Another type of general production var-iable is the method used for construction; which may be a companyphilosophy or a constraint of the available equipment.

Time production variables change only with respect to time. Anexample of a time production variable is the number of holidays per

month. It should be expected that production will be lower duringDecember than August solely on the basis of the holiday season inDecember.

Conditions that change with respect to horizontal location alongthe alignment are location production variables. Examples of suchchanges include terrain, site conditions, geotechnical conditions(e.g., existence of rock), urbanization, or right-of-way width alongthe project. These variables require the scheduler to change the pro-duction rates with respect to locations along the horizontal align-ment. For example, one can visualize the variation in constructionconditions when constructing pipeline in the mountains versus flatprairie land.

The last type of variable, time-location production variables,change with respect to time and location. Examples of these pro-duction variables include weather and environmental windows. Forexample, performing construction during the winter months is typ-ically more difficult than during the summer months. However,weather is also dependent upon location because the winter inWyoming is quite different than winter in Florida. A linear projectoften spans a time and distance great enough to see these types ofvariation in weather patterns. Another example of a productionvariable that changes with time and location is environmentalwindows. For example, an environmental window may negateall construction during the months of March through July for acertain location because of wildlife constraints.

Determining the variables and types of the variables that affectthe activities of a given construction project is the foundation forcreating a linear schedule with changing production rates. Thevariables themselves dictate the magnitude of the changes in pro-duction rate, and the type of production variable determines howthe changing rates will be applied.

Linear Scheduling Model with Varying ProductionRates

In this study, the linear scheduling model with varying productionrates (LSMVPR) has been developed as a framework for applyingchanges in production rates when and where they occur in time andspace for a given linear construction project. LSMVPR creates thisframework based on the concept of the working window (WW),which is defined in this paragraph. A traditional linear scheduledepicts the entire time and location when and where the construc-tion is proposed. The overall time-location for the entire project isreferred in this research as the project’s time-location chart (TLC).For the purposes of this paper, the TLC is assumed to depict time onthe ordinate and location on the abscissa. When dealing with fac-tors that affect production rates, it is necessary to look at smallerpieces of the TLC. When the TLC is sliced into a grid of smallercells on a user-defined interval, these cells depict the project’sworking windows. AWW is a time-space rectangle with a homog-enous set of variables that affect the construction production rate.The next section further discusses the concept of working windows.

Working Windows

Working windows display when and where the production varia-bles may change along the linear project. Working windows areareas of time and location for which unique production variablescan be assigned (e.g., a given working window has an average slopeof 0.01). Because a linear scheduling chart depicts time on one axisand location on the other axis, drawing a grid on this chart breaksthe chart into areas of time and location. Fig. 2 is a general view of agrid of working windows that split up a project. The nomenclature

Table 1. Types of Production Variables with Examples

Type of ProductionVariable Examples

General Number of workers, safety requirements,

construction methods

Time Work week, holiday schedule, learning curve

Location Terrain, urbanization, site conditions, geotechnical

data, work space

Time-Location Weather, environmental windows, site conditions



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for working windows is given as WWij, where i = column andj = row. The appropriate production variables that should be appliedto a working window can be determined by looking up the i and jcoordinates for its working window.

Fig. 3 depicts a more detailed view of a typical working window.The working window naming convention shown in Fig. 3 applies towork moving from left to right or from lower stationing to higherstationing along the horizontal alignment. The location of thewindow begins with the working window location start (WWLS)and ends with the working window location end (WWLE). Corre-sponding nomenclature depicts the time start and end with Workingwindow time start (WWTS) andworkingwindow time end (WWTE),respectively. Again the use of the i and j variables allow a uniqueidentifier for each working window, and the corresponding variablecarries through when naming the start and end of each window.

Fig. 4 adds an activity to the working window along withnomenclature to specify the entry and exit coordinates of theactivity. All activities will move in a straight line through the work-ing window because by definition the working window’s produc-tion variables are constant, and thus the production rate through thewindow is constant. The nomenclature for naming the coordinatesof the activity vertices as it moves through the chart is to start at Xn,Yn, and move to Xnþ1, Ynþ1. X represents the distance or stationingcoordinate, and Y represents the time coordinate. The subscript n isthe number of the vertex as the activity enters the working window,and the subscript nþ 1 denotes the coordinate of the vertex as theactivity exits the working window. The vertices are numbered fromleft to right with the start of the activity beginning with the numberzero, or X0, Y0. These vertices exist at every change in the workingwindow even if the activity does not change slope through the

working window. All calculations performed are based on activitymovement from left to right, with time increasing from bottomto top.

LSMVPR Calculations

Fig. 4 also includes additional terminology in the diagram to depictinformation necessary for making calculations for developing alinear schedule. The distance remaining (DR) is the amount of dis-tance that has not been completed in the current working windowwhen the activity starts in that window. The time remaining (TR) isthe amount of time that is remaining in the current working windowwhen the activity starts in that window. Distance remaining andtime remaining can be calculated with the following equations:

DRij ¼ WWLEi � Xn ð1Þ

TRij ¼ WWTEj � Yn ð2ÞDistance remaining and time remaining are used to determine

the movement of the activity through the linear scheduling chart;the movement from working window to working window. For ex-ample, there are three locations the activity can exit the workingwindow once it enters: it can cross the top time axis, the right dis-tance axis, or it can exit at the intersection of the two. The exitlocation is determined by a combination of the DR, TR, and pro-duction rate for that working window. A variable called distancetraveled in time remaining (DTTR) is introduced for determiningthe exit location. Eq. (3) is the equation for DTTR, where PRij =production rate for the given working window:

DTTRij ¼ PRij � TRij ð3ÞThe DTTR can then be compared with the DR to determine the

exit location. The following three outcomes can occur:1. DTTRij ¼ DRij → Activity exits at the intersection of the

top time axis and right distance axis of the working window(Fig. 5);

2. DTTRij > DRij → Activity exits at the right distance axis ofthe working window (Fig. 6); or

3. DTTRij < DRij → Activity exits at the top time axis of theworking window (Fig. 7).

Figs. 5–7 graphically illustrate the three cases.The next production rate can be determined by understanding

how the activity exits the working window. Fig. 8 illustrates thecases in which the activity enters and exits the working window.Again, all examples and calculations in LSMVPR are based on

Fig. 2. Naming conventions for working windows on time-locationchart

Fig. 3. Individual working window nomenclature Fig. 4. Activity and working window nomenclature



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working left to right across the chart, with location along the x-axisand time along the y-axis. The first row of examples is indicative ofthe activity entering the working window along the time start axis,and the second row illustrates activities that enter along the distancestart axis. The third row depicts activities that enter the workingwindow at the intersection of the time start and distance start axes.The figures are further grouped by the exit location, which is thedistance end axis, intersection of the distance end axis and time endaxis, time end axis, and time end axis for columns 1, 2, 3, and 4,respectively. Column 4 depicts a special condition of exitingthrough the time end axis, in which case the production rate forthe working window is equal to zero because of a nonworking day.

Activity Performance Index

The visual nature of linear scheduling can be enhanced by addingcolor to the individual working windows to provide the userwith additional information about the production rates predictedwithin each working window. Displaying color in the workingwindows indicates calculated performance relative to the desiredproduction rate. The areas difficult for construction can then beeasily determined.

The color added to the working windows is referred to as theactivity performance index (API). The API is a color scheme thatindicates the status of production rates on the project. For example,red indicates very poor performance, and green indicates favorableperformance with regard to the desired production rate. The colorindicates the relationship between a user-defined production rate(PRUD) and the calculated production rate, which is a most likelyrate based on historical data (derived from regression equations and

using LSMVPR). Table 2 illustrates the default percentages forassigning color based on production performance.

To derive the percentages, the user determines a level of produc-tion desired for the given activity. At every working window, aproduction rate is determined based on LSMVPR. The percentageis then determined for each working window on the basis ofEq. (4). The color indicates the calculated production rate dividedby the user-defined production rate, as shown in Eq. (4):

APIij ¼ PRij=PRUD � 100 ð4Þ

For example, if the scheduler desires a production rate of 10,000linear feet per day for a given activity, but the calculated productionrate for the given working window is 8,500, the API ¼ 85%. Thisindicates the predicted production rate for that activity in that work-ing window is 85% of the desired production level, thus the work-ing window is shaded blue. This visual aid helps the schedulereasily determine the time-locations that may be problematic forconstruction. For instance, if the project requires welding to moveat a rate of 10,000 linear feet per day, but the calculated productionrate is less than 5,000 feet per day, the user can easily see the redworking windows indicating that historically this production ratehas not been achieved under the given conditions. This patternof color also aids in determining optimal starting locations anddates for the crews along the horizontal alignment, providing avaluable front-end planning tool.

The color shaded in the working window is activity-dependentby definition. This means the same working window could be dif-ferent colors based on the calculated API for the different activities,so it is important to keep in mind which activity or activities drivethe work. It is more important to portray the obstacles with the driv-ing activity because the work is planned around the driver(s). Each

Fig. 5. Case 1: DTTR is equal to DR

Fig. 6. Case 2: DTTR is greater than DR

Fig. 7. Case 3: DTTR is less than DR



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activity may be affected by different variables, thus each activityhas its own API chart. The API chart depicted on the finished linearschedule is typically the driving activity for the construction effort.For instance, for natural gas pipeline construction, welding is themost important activity, thus the schedule should be presented withthe API based on the values for the welding activity.

LSMVPR Algorithm

Fig. 9 shows an overview of the calculation procedure for LSMVPR.The algorithm developed to calculate variable production rate linearschedules is based on a forward- and backward-pass methodology.In general, the forward pass schedules the activity using the mini-mum lead (ML) specified from the activity input stage. The ML isthe minimum separation between activities based on time units. Forexample, Activity A may require a 10-day start ahead of Activity Bto keep the crews for the respective activities from interrupting oneanother’s work. This 10-day buffer is the ML and corresponds to astart to start relationship in CPM. For the initial calculation, theactivity separation (AS) is set to the ML. The AS is the differencebetween the start of the preceding activity and the activity beingscheduled.

A backward pass is then performed to ensure that ML is satisfiedthroughout the length of the activity. During the backward pass, the

time difference between every vertex of both the activity beingscheduled and the preceding activity is calculated. The least timeinterval (LTI) is the minimum separation of time calculated be-tween the two activities. The LTI is then compared to the ML.If the LTI is greater than or equal to the ML, the next activitycan be scheduled. If the LTI is less than the ML, the AS is increasedby a value equal to the time iteration interval (TII). The TII is a user-defined time interval. This process creates an iterative loop until theLTI is greater than or equal to the ML. This looping nature is nec-essary to ensure the Minimum Lead is satisfied because of the pos-sibility of incurring varying production rates with each iteration.

The steps to construct a linear schedule using the LSMVPR oncethe initial data has been entered are as follows:1. Set the start date for the first activity to the project start date

and subsequent activities to a start date equal to the start of thepredecessor plus the ML required.

2. Set the AS equal to the ML of the preceding activity, zero if nopredecessor exists, as in the case of the first activity in theschedule.

3. Lookup the production variables for the current workingwindow (WWij).

4. Calculate the production rate for the current working window.5. Calculate distance remaining (DRij), time remaining (TRij),

and distance traveled in remaining time (DTTRij).6. Use the following criteria to determine the exit location for the

activity from the current working window:a. DTTRij ¼ DRij → Activity exits at the intersection of

the top time axis and right distance axis of the workingwindow.

b. DTTRij > DRij → Activity exits at the right distance axisof the working window.

c. DTTRij < DRij → Activity exits at the top time axis of theworking window.

7. Use the following criteria to calculate the exit coordinate forthe activity:

Fig. 8. Cases for the entry and exit of working windows by an activity

Table 2. Activity Performance Index and Corresponding Default ColorScheme

Upper Lower Color

100% or greater 90% Green

89% 80% Blue

79% 70% Yellow

69% 60% Orange

59% 50% or less Red



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a. DTTRij ¼ DRij → ðWWLEi;WWTEjÞ ð5Þ

b. DTTRij > DRij → fWWLEi; ½ðDRij=PRijÞ þ Yn�g ð6Þc. DTTRij < DRij → ½ðXn þ DTTRijÞ;WWTEj� ð7Þ

The naming convention for the exit coordinate follows (X, Y),where X represents location or distance on the project, and Y rep-resents time.8. Determine if the activity has been calculated to the end of the

project.9. If not, go to the next working window, and repeat Steps 3–8

until the activity reaches the end of the project.10. Calculate the time difference vertically for every vertice in the

current activity and the predecessor.11. Set the minimum distance value from Step 10 to the LTI.12. Compare the LTI with the ML required, and determine if the

LTI is greater than or equal to the ML required.13. If not, set the AS to the AS plus the TII, and repeat Steps 3–12

until the LTI is greater than or equal to the ML required.14. Next activity, repeat Steps 1–13 for all activities.15. Linear schedule complete.

Application of LSMVPR

A simple example project is used to show how LSMVPR works. Thefollowing example is a generic linear construction project consist-ing of three activities. The three activities are affected by differentvariables and have generic regression equations to predict their pro-duction rate. Assume the following:

The sequence of construction is Activity A, then Activity B,followed by Activity C. The activities have a production rate basedon the generic regression equations below:

PRA ¼ WD � 100PRB ¼ WD � ð50� 0:33 � LV1Þ

PRC ¼ WD � ð100� 0:5 � TLV1Þwhere PRA, PRB, PRC = production rates for Activities A, B,and C, respectively; WD = working day (0 = not a working day,1 = working day), LV1 = location variable 1, TLV1 = time-locationvariable 1

The variables relating to production rate affect the constructionproject as follows:

Fig. 9. Overview of the calculation procedure for LSMVPR



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Time Period 1 has a no-work zone for all activities from location50–150;LV1 has a value of 30 from location 150–250 and 0 for all otherlocations; andTLV1 has a value of 100 from location 0–150 for time periods 4–8and 0 for all other time locations.The activities have desired production rates of 100, 50, and 100 forActivities A, B, and C, respectively. The required time buffer be-tween the activities is 0.5 time periods. The construction will startat location 0 and time period 0 and progress to the right and top ofthe time-location chart.

The API charts for the activities are depicted in Figs. 10–12 forActivities A, B, and C, respectively.

Each activity’s API chart depicts the corresponding change inAPI caused by the variables affecting that activity’s production rate.To determine the color displayed on the API chart, the predictedproduction rate based on the regression equations is divided bythe user’s desired production rate. This value is converted to a per-centage and compared to the API color assignment to determine thecolor displayed on the chart. For example, Activity C is influencedby TLV1 with a predicted production rate of 50 units per unit timein the time-location of time equal to 4–8 and location equal to0–150. This production rate is half or 50% of the desired produc-tion rate. The corresponding color on the API chart for 50% of thedesired production rate is blue, thus the region described appears inblue (Fig. 12).

Figs. 13–15 show sequential development of linear schedulesfor the activities on the API charts. Fig. 15 depicts the final linearschedule for the three activities with the API chart for Activity C.Because the API chart is different for each activity based on the

degree the variables affect each activity, it may be necessary to re-view the final linear schedule with different API charts shown. Forexample, the API chart in Fig. 15 shows when and where changesare predicted for Activity C. It would be necessary for the user toreference the API chart for Activity B to determine the regions inwhich the production rate changes for Activity B. ReviewingFig. 15 helps the user understand that Activity B’s production ratewill fall in the location from 150–250 regardless of the time atwhich Activity B is performed in this area. This example illustratesLSMVPR’s ability to convey more information to the user thanprevious linear scheduling models. The user now has the abilityto visualize slow areas of construction based on any number ofvariables and plan accordingly. The user also has a scheduling tool

Fig. 10. (Color) API chart for Activity A

Fig. 11. (Color) API chart for Activity B

Fig. 12. (Color) API chart for Activity C

Fig. 13. (Color) Activity A shown on the API chart for Activity A

Fig. 14. (Color) Activities A and B shown on the API chart forActivity B



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that more accurately illustrates the expected production rate of eachactivity for any time-location window in the schedule. LSMVPRprovides the framework for quantifying advantageous timingand crew distribution to complete linear projects while avoidingthe most difficult time-location windows.

Conclusions

This research showed that the changes in production rates causedby to time and location can be modeled for use in predicting futureconstruction projects. The model created for this purpose isLSMVPR. Using LSMVPR allows the scheduler to develop scheduledurations based on minimal project information. The model alsoallows the scheduler to analyze the impact of various routes or startdates for construction and the corresponding impact on the sched-ule. The graphical format also allows the construction team to visu-alize the obstacles in the project when and where they occur byusing a new feature called the API. This index is used to colorthe linear scheduling chart by time and location, with the variationin color indicating the variance in predicted production rate fromthe desired production rate.

This research has laid a foundation for developing linear sched-ules that take into account varying productions rates when andwhere they occur. Further research could expand upon three majorareas: (1) data collected for additional site specific or projectspecific considerations, (2) expanding the capability of LSMVPRto include additional features, and (3) expanding the data collectedto include other types of linear projects.

Expanding the abilities of LSMVPR would also aid in the analy-sis of complex linear construction projects. Additional features thatwould improve the capabilities include (1) allowing the ability touse multiple crews starting in multiple locations, (2) the ability tomodel activities moving across the project in both directions,(3) incorporating nonlinear activities into the scheduling model,(4) including additional activity types, and (5) incorporating

Bayesian updating methods to allow updates to the production ratemodel while construction is in progress.

Finally, the model could be applied to other types of linearprojects. The framework developed can be applied to almost anylinear project. Expanding the range of linear projects would requirecollecting data corresponding to the activities in those projects.

References

Chao, L., and Skibniewski, M. J. (1994). “Estimating construction produc-tivity: Neural-network-based approach.” J. Comput. Civ. Eng., 8(2),234–251.

Chong, W. K. (2005). “Construction production rate information system forhighway projects.” Ph.D. thesis, Univ. of Texas, Austin, TX.

Chrzanowski, E. N., and Johnston, D. W. (1986). “Application of linearscheduling.” J. Constr. Eng. Manage., 112(4), 476–491.

Cosma, C. (2003). “Development of a systematic approach for uncertaintyassessment in scheduling multiple repetitive construction processes byusing fuzzy set theory.” Ph.D. thesis, Univ. of Florida, Gainesville, FL.

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Harmelink, D. J. (1995). “Linear scheduling model: The development of aliner scheduling model with micro computer applications of highwayconstruction control.” Ph.D. thesis, Iowa State Univ., Ames, IA.

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Herbsman, Z. J. (1999). “The application of linear scheduling in the FDOTconstruction operation.” WPI #0510851, Florida Dept. of Transporta-tion, Gainesville, FL.

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Johnston, D. W. (1981). “Linear scheduling method for highway construc-tion.” J. Const. Div., 107(2), 247–261.

Kuo, Y. (2004). “Highway earthwork and pavement production rates forconstruction time estimation.” Ph.D. thesis, Univ. of Texas, Austin, TX.

Liu, S. (1999). “Resource-constrained scheduling of linear projects: A heu-ristic approach using tabu search heuristics.” Ph.D. thesis, Purdue Univ.,Lafayette, IN.

Mattila, K. G. (1997). “Resource leveling of linear schedules: A mathemati-cal approach using integer linear programming.” Ph.D. thesis, PurdueUniv., Lafayette, IN.

O’Connor, J., and Huh, Y. (2005). “Crew production rates for contract timeestimation: Bent footing, column, and cap of highway bridges.”J. Constr. Eng. Manage., 131(9), 1013–1020.

Smith, S. D. (1999). “Earthmoving productivity estimation using linear re-gression techniques.” J. Constr. Eng. Manage., 125(3), 0133–0141.

Vorster, M. C., Beliveau, Y. J., and Bafna, T. (1992). “Linear schedulingand visualization.” Transportation Research Record 1351, Transporta-tion Research Board, Washington, DC, 32–39.

Yamin, R. A. (2001). “Cumulative effect of productivity rates in linearschedules.” Ph.D. thesis, Purdue Univ., Lafayette, IN.

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Fig. 15. (Color) Activities A, B, and C shown on the API chart forActivity C



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http://dx.doi.org/10.1061/(ASCE)0887-3801(1994)8:2(234)





http://dx.doi.org/10.1080/15578770701429431



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